For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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1answer
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Co-ordinate Geometry Equation of a circle

Find the equation of the circle having the lines x+1=0 and x-3=0 as tangents and with its center lying on the line y=3 I don't know much maths so if you could tell me also how to get the co-ordinates ...
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0answers
32 views

Prove that the figure is a trapezoid…

Knowing that triangle $ABC$ is an arbitrary one, points $D$,$F$,$G$ are midpoints of respective sides of the triangle(as you see in the picture ), and $CE$ is the altitude, prove that $DG$ is ...
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2answers
35 views

Does the median make angles in the same proportion as the sides?

Till I remember I had studied this in the lower classes, but am not sure whether this is true or not. In the figure CD is a median. Does CD divide the angles 1 and 2 in the same ratio of the sides a ...
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2answers
27 views

Prove that the measure of the angle…

How can I prove that the measure of angle $EBC$ is $60$? Thank you!
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1answer
17 views

How do I justify that a second order cone is an intersection of half space

I am studying convex optimization right now, and the text book claims that a second order cone is a collection of intersections of half space $$K_n = \bigcap_{ u:\|u\|_2 \leq 1} \{(x,y) \in R^{n+1 } ...
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1answer
26 views

Formula for greatest cross section of regular dodecahedron.

Is there a formula for the area of greatest cross section of a regular dodecahedron? For example, a hole big enough for it to fit into.
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0answers
19 views

How to test for a polygon witn n vertices if it's nonintersecting polygon or not?

How can you design an algorithm to know if an n-vertex polygon nonintersecting ? On what criteria is the test going to be
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0answers
21 views

How to calculate normal (of magnitude 1) of a triangle?

I am currently doing a bit of geometry practice and wanted to know how to calculate the normal (of magnitude 1) of a triangle defined by 3 vertices: a, b and c`. ...
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2answers
29 views

Two triangles, only one different side, same area?

Suppose the following triangles: Where BC = CD. Obviously, the area of ABC and ACD are equal, since they both share the same base, and the same height, namely, AB. I was able to prove that their ...
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1answer
21 views

Prove if one side of a triangle is a common measure of the other two sides, then the triangle is isosceles.

The definition of a common measure in my text book is this : A common measure of two segments is a third segment such that it is contained in each of the first two a whole number of times with no ...
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2answers
67 views

Finding the measure of an arc on a circle

If someone could work me through how to solve this, that would be great because I am stumped on this one. I know it looks like there is a lot of useless information in the picture, but there are ...
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0answers
28 views

Isometry in Euclidean space

The question is to show that an isometry from $\mathbb{E}^{1} \to \mathbb{E}^{1}$ is of the form $x \to ax + b $ from first principles, and determine the values $a$ can take. From my notes I know for ...
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0answers
17 views

Finding the equation of a plane by using point-to-point distances

Assume that we have a plane $P_1$ whose equation is known. I need to find the equation of plane $P_2$. If we choose a point set $N = \{n_1, n_2, ...\}$ on $P_1$ and another point set $M = \{m_1, m_2, ...
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2answers
43 views

Provide a proof for the following problem…

How can I prove from this image that $BQ=2*PE.$ We know that $TM$ is parallel to $QD$ and that $CF$ is the bisector of angle $C$. As you see $QD$ and $TM$ are perpendicular to $CF$. Obiously I found ...
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2answers
33 views

Is the rhombic dodecahedron the only isohedral polyhedron that tiles 3-space (other than the cube)?

Is the rhombic dodecahedron the only face-transitive (or isohedral, i.e. all faces are the same) polyhedron that seamlessly tiles 3-dimensional Euclidean space (other than the cube)? I'm looking ...
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2answers
20 views

Proof required for the following problem…

Can you please give an explanation of how can I find that $K$ and $P$ are midpoints of respectively $CR$ and $CS$. NOTE! that: $AE$ and $DB$ are bisectors of angles $A$ and $B$. and $CK$ and $CP$ ...
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1answer
41 views

A quadrilateral ABCD satisfies the following relationship with respect to any point $M$ in the plane $AM^2+ CM^2 = BM^2 + DM^2$

This quadrilateral could be A parallelogram A rectangle A square A rhombus None of the above The answer can have multiple answers. Please provide the proof of your answer. I would be grateful ...
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2answers
21 views

Multiplying a vector by a constat

This is probably a very silly question, but I just can't remember... If vector u=-8i+32j, how can I multiply it with a constant a? Would the new vector be: u=-8ai+32j ???
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1answer
44 views

Find the length of…

Find the length of $\overline{AD}$ knowing that it is divided into three equal parts by to tangent circles with radius respectively $3\sqrt{3}$ and $\sqrt{3}$ . Here's the graph: so the segments ...
2
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1answer
40 views

A group acts on a disk

Let $G$ be a group, it acts continuously on a disk $\mathbb{D}$, $g$ is a non-trivial central element of $G$. The set of fixed points of $g$ is $\partial\mathbb{D}$, I want to prove that for $h\in G$, ...
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1answer
32 views

Transformation of ellipsoid to sphere

So I need to find an volume-preservating mapping from an ellipsoid to a ball (solid sphere). (Specifically: x^2/9 + y^2 + z^2 <= 3, but I'd rather understand the general case than just get how to ...
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1answer
43 views

The area of a region around a curve

If we are given a simple closed curve with length $L$ in the plane, and we have a fixed number $r$ such that for each point $x$ on the curve there is a related disc $D(x,r)$ with radius $r$, how can ...
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0answers
29 views

Determining how accurate an ellipse fit is

So I have an image of bacteria particles which are often shaped very irregularly with many grooves. Im trying to fit ellipses onto these particles so I can get a better, more smooth analysis of the ...
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0answers
56 views

Escaping from the attack dog [duplicate]

Suppose you're at the center of a circular field of radius 100 ft and there is a guard dog on the perimeter of the field. If the guard dog runs 4 times faster than you but cannot leave the perimeter, ...
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1answer
36 views

Locate a point a given distance from another point on an ellipse

Similar to Point on circumference a given distance from another point, but for an ellipse. Unfortunately, the difference is non-trivial. I have an ellipse and a point (C) that is somewhere on the ...
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1answer
24 views

fixed length random chord outside of circle.

consider a uniform distribution on a unit circle, I construct a cord by the following steps: pick one endpoint A within the unit circle uniformly. points that are $0<d<1$ distance away from ...
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1answer
29 views

Area of intersection between line and circle?

I have a circle $\mathcal{C}$ and a line $\mathcal{L}$ in the euclidean plane. Let say that the equation of the circle and the line are given respectively by: $$E_{\mathcal{C}}: ...
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1answer
34 views

area enclosed within 3 lines

Equations of lines $L1$ and $L2$ are $y = x − 2$ and $y = −2x − 2$. If $y = −x$ is the angle bisector of lines $L2$ and $L3$, then what is the area enclosed within the 3 lines $L1, L2$ and $L3$? ...
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1answer
38 views

The relation between the radiuses…

Find $\frac{R}{r}$ where $R$ is the radius of the circumscribed circle of a trapezoid and $r$ is the radius of the inscribed circle of this trapezoid. Thank you!
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3answers
68 views

Perpendicular lines inside and outside a circle

No trigonometry allowed. Let $\Delta ABC$ be inscribed inside a circle.Let $P$ be a point on the circle.Let $PD$ and $PE$ be perpendiculars on on $BC$ and $AC$ respectively.Let $DE$ when extended ...
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1answer
84 views

if $AD=BD,\angle ADC=3\angle CAB,AB=\sqrt{2},BC=\sqrt{17},CD=\sqrt{10}$,How find $AC$

in quadrilateral $ABCD$,such $$AD=BD,\angle ADC=3\angle CAB,AB=\sqrt{2},BC=\sqrt{17},CD=\sqrt{10}$$ Find the $AC=?$ My idea: let $$\angle CAB=x.\angle ADC=3x,\angle ADB=y,$$ then we have $$\angle ...
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1answer
56 views

Problem concerning inscribed and circumscribed circles…

Can you please help me solve this really difficult problem: Find R/r where R is the radius of the circumscribed circle of a trapezoid and r is the radius of the inscribed circle of this trapezoid. ...
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0answers
34 views

Chern class of line bundle and vector bundle

Let $L$ is a Line bundle and $E$ a vector bundle of rank $r$ then how can we prove that $$c_1(L\otimes E)=rc_1(L)+c_1(E)$$ where here $c_1$ means first chern class
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0answers
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Suppose you have a polygon with $n$ vertices. How many triangulations are possible, when a 'hole' is in the middle?

Given a polygon with $n$ vertices. We know the answer to the questions "How many triangulations are there?" is Catalan numbers. However, I wish to consider a variant of this case. Suppose still that ...
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2answers
28 views

How do we define touching lines?

If two curves are touching at one point and intersect one another, how do we define it? If two lines are touching at a point then $L\cap K=\{q\}$ for two lines L and K and q is the touching point. ...
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1answer
42 views

Help with this trigonometry problem?

Is there an easier way of doing this problem: A square tower stands upon a horizontal plane. From a point in this place from which three of its upper corners are visible their angular elevations ...
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2answers
59 views

What is the most elementary but still correct according to the most rigorous standard proof of the isoperimetric inequality?

Can you write the most elementary proof of the isoperimetric inequality (but still correct according to the most rigorous standard )? $$l^2> 4πA$$
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1answer
21 views

A mapping that maps circles centered at origin to lines parallel the real axis

I need a mapping that maps a circle (with the center at the origin) to lines parallel to real axis:
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0answers
38 views

Is $\nabla_{X}:\Gamma(E) \to \Gamma(E)$ continuos aplication?

Be $\Gamma(E)$ smooth sections space of an vector bundle, $\Gamma(E)$ with Fréchet topology. Gived a conexion in E and a smooth vector field $X$, Is $\nabla_{X}: \Gamma(E) \to \Gamma(E)$ continuos ...
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0answers
34 views

Polygones inscribed with in a circle

Let's say that there is a circle in two dimension and the diameter of the circle is 1.First start with an equilateral triangle inscribed with in the circle and the measure of the angles are equal to ...
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2answers
57 views

Finding the lengths of the sides of a triangle given 3 angles only.

If a right triangle ABC with an angle A at 90 degree, B 45 degree, C 45 degree is their a way of finding the length of the sides abc without knowing any of their lengths. Normally we use ...
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1answer
29 views

Why does the sign of ax+by+c change, for the coordinates of a point that is not on the line?

So, all the points (x,y) satisfying ax+by+c = 0, lie on the staright line. But if a point is above or below that line, the sign of the function f(x,y) = ax+by+c, changes, being negative or positive, ...
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1answer
29 views

Convex Sets and extreme supports

Let the set $S$ in $R^n$ consists of the origin $0$ and $n$ lineary independent vectors $T_1, \ldots, T_n$. Show that $C(S)$, the convex hull of of $S$, is the intersection of its extreme supports, ...
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0answers
86 views

Construct a line which intersects the interior two circles at chords of equal length.

I am stuck on a geometry construction and proof. Construct a line which intersects two circles at chords of equal length. You are given two circles (center point included) of different sizes and ...
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0answers
26 views

Right triangles, minimum difference of two measured angles

The question reads: If $\angle ABC$ is a right angle and the measure of $\mu(\angle ACB)=r^{\circ}$, what is the minimum difference between $\mu(\angle ACD)$ and $\mu(\angle BAC)$? Which of the three ...
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6answers
4k views

Area covered by a constant length segment rotating around the center of a square.

This is an idea I have had in my head for years and years and I would like to know the answer, and also I would like to know if it's somehow relevant to anything or useless. I describe my thoughts ...
0
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1answer
29 views

Find a vector non-orthogonal to a given set

Let $S = \{\mathbf{v}_1, \dots, \mathbf{v}_n\}$ be a set of vectors in $\mathbb{R}^{n}$. I would like to find a vector $\mathbf{u} \in \mathbb{R}^n$ such that, for all $i \in [1, n]$, $\mathbf{u}$ and ...
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1answer
34 views

Geometric reasoning and triangular coordinates

The following is from a book: I do not understand the sentence "... the point $(t_1, t_2, t_3)$ can be plotted by plotting $(t_1 = t_3, t_2 = t_3)$...", what is meant by the point $(t_1 = t_3, t_2 = ...
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1answer
41 views

Problem plotting hypotrochoids using a computer

I have been trying to use a computer to plot some hypotrochoids, but I've run into some issues. For those that are unfamiliar, the parametric equations of a hypotrochoid are: $$x(\theta) = (R - ...
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Transformation matrix from quadrilateral to rectangle

There exists a rectangle somewhere in space with some orientation. A camera from the coordinate center point is looking along the z axis and is seeing the rectangle as a quadrilateral (due to ...